Assume that our universe is a mathematical one, similar to the one that Tegmark proposed (see here). In contrast to what I read there, let's assume that the axioms upon we build the universe are such that they lead, according to Gödel, to statements in our physical universe that can't be proven i.e. measured, like e.g. position and velocity of a quantum particle, according to Heisenberg's uncertainity principle.

Are there modern philosophers that tried to build a connection between the free will of the conscious mind and Gödel's incompleteness in a mathematical universe?

You are mixing up issues from three different realms: 1) the problem of free will, a question for neuroscience. 2) Gödel's incompleteness theorem, a mathematical theorem about the non-provability of certain true statements in a consistent formal theory of sufficient strenght 3) The idea to consider our physical world as a mathematical simulation by a digital computer. But you don't elaborate how the three issues could be related, e.g., how to set in relation no. 2 with Heisenberg's uncertainty principle in the physical world of quantum mechanics. Could you please add some elaboration, thanks.
– Jo WehlerMar 14 '16 at 22:09

@JoWehler (i) a particle always has a definite position and velocity, but we can't prove, i.e. measure that, due to Heisenberg's uncertainty principle, so the HUP could be a consequence of GIT. (ii) I feared that "free will" might lead to irritations. Let me try to improve on this... (iii) I never thought of the physical world as a simulation...
– draks ...Mar 14 '16 at 22:30

ad i) But why relating non-provability to being not-simultaneously-measurable? - ad iii) The core issue of a mathematical universe is reasoning about the physical world as a mathematical simulation.
– Jo WehlerMar 14 '16 at 22:39

@JoWehler re(i) it just an example, the impossibility to construct a detector that reveals the particle character of gravitons (because it would be so heavy that it collapse to a black hole) is another... re(iii) then my mathematical universe is different...
– draks ...Mar 14 '16 at 23:01

Hawking published an essay online some years back arguing from incompleteness that physics cannot be completed. For reasons unknown he has since taken it down.
– PeterJMar 6 at 16:10

2 Answers
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let's assume that the axioms upon we build the universe are such that they lead, according to Gödel, to statements in our physical univerese that can't to proven i.e. measured, like e.g. position and velocity of a quantum particle, according to Heisenberg's uncertainity principle.

This is not accurate. Gödel's theorem is about formal languages (which are abstract mathematical structures) and has nothing to do with the physical universe directly. It is not related at all to Heisenberg's uncertainty principle.

That being said, there have been philosophers who have tried to establish a connection between freewill and consciousness on one hand, and Gödel's theorem on the other. In particular Lucas (Lucas, J. R. (1961). “Minds, Machines and Gödel,” Philosophy 36:112-127.) argued that human consciousness is different from machine intelligence for its ability to recognize the truth of a Gödel sentence, while a machine can never do so using any algorithmic process.

Closer to your idea of their being a connection between Gödel's theorem, consciousness and physics, Penrose presents a modern version of Lucas' argument, which he takes on step further by infering a connection between the Gödel's theorem based argument and quantum mechanics (Penrose, R. (1994). Shadows of the Mind. Oxford: Oxford University Press. p 395). In particular, he posits that this additional ability that human minds have over computers stems from quantum mechanical phenomena happening in the brain.

Penrose, together with biologist Stuart Hamerhoff, has developed the Orch-Or model of how consciousness arises at the quantum level inside individual neurons, as opposed from the connections between networks of neurons (Hameroff, S; Penrose, R (March 2014). "Consciousness in the universe: A review of the ‘Orch OR’ theory". Physics of Life Reviews (Elsevier) 11 (1): 39–78.)

Max Tegmark disagrees with Hamerhoff and Penrose, and thinks that their arguments about consciousness being based on quantum mechanical principles (Tegmark, M., 2000, “Importance of quantum decoherence in brain processes,” Physical Review E 61, 4194–4206.) are mistaken. Roughly speaking he thinks that the brain is "too warm" for it quantum states to last long enough to be the source of consciousness. His objections are purely physical and are not related to the Gödel/mathematical side of Penrose's theory.

+1 and Thanks for plenty of references, but I feel misunderstood: When the universe is mathematical, i.e. based on some axioms, then according to Gödel there are unprovable statements within this formal language and therefore within this universe. Heisenberg just served as an example. Maybe my point isn't very clear, and if you could help me improving it I'd be very glad...
– draks ...Mar 14 '16 at 21:45

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I understand why you might think that Godel and Heisenberg are related. But physical measurements and provability are two different concepts. You seem to be indicating that they are the same and that makes the question a little confusing. Especially the following "statements in our physical univerese that can't to proven i.e. measured,...". Statements are never in the physical universe, at most, they are only about the physical universe, and Godel's limit applies to any formal theory, regardless of whether it is about physical entities or abstract entities.
– Alexander S KingMar 15 '16 at 4:57

@AlexanderSKing Your opening is basically equivalent to 'Tegmark is crazy', from which point it becomes impossible to answer the question. The hypothesis is that the universe itself is a mathematical object. It is not that science is a mathematical object about modelling the universe. It is that the universe itself is a mathematical structure, and science approximates it with another one.
– jobermarkMar 6 at 2:20

It appears to me that what Tegmark specifically proposes is not readily intelligible, see How can the physical world be an abstract mathematical structure? Even if we make it more intelligible by adjoining a God-like entity that animates symbology into reality Gödel's incompleteness is essentially inconsequential for Tegmark because he is a physical Platonist. I.e. for him mathematical truths about reality exist out there regardless of whether or not our first order languages happen to be adequate for proving them. At best, one can get a compatibilist account of free will out of it, where, although everything is mathematically predetermined, some things are transparently so, because they are "provable" in our theories, and others are only transcendentally so in the Platonic sense.

But, as our formal capabilities develop, the latter may move into the former column, after all, the original Gödel sentences become provable when the original theory is strengthened. So as we advance we are destined to find out more and more how what seemed "free" before was in fact a mathematical necessity. But this is a common theme of compatibilism whether or not incompleteness is invoked to back it up.

The most famous attempt to apply Gödel's incompleteness ideas to explaining puzzles of conscious mind are Hofstadter's classic Gödel, Escher, Bach, and its sequel I Am a Strange Loop. The added bonus is that Hofstadter discusses in detail the incompleteness based arguments of Gödel, Lucas and others for human creativity making a qualitative difference between the man and the machine. Here is Lucas:

"However complicated a machine we construct, it will, if it is a machine, correspond to a formal system, which in tum will be liable to the Gödel procedure for finding a formula unprovable-in-that-system. This formula the machine will be unable to produce as being true, although a mind can see it is true... In a sense, just because the mind has the last word, it can always pick a hole in any formal system presented to it as a model of its own workings. The mechanical model must be, in some sense, finite and definite: and then the mind can always go one better".

Alas, this argument is overly optimistic about mind's capabilities. According to Hofstadter himself, it is self-reference ("loop") inherent in Gödel's construction that allows computing and processing to imbue itself with meaning and understanding, and therefore "create" consciousness and "I". Martin Gardner writes in his review of I Am a Strange Loop:

"Consciousness for Hofstadter is an illusion, along with free will, although both are unavoidable, powerful mirages. We feel as if a self is hiding inside our skull, but it is an illusion made up of millions of little loops. In a footnote on page 374 he likens the soul to a “swarm of colored butterflies fluttering in an orchard.” Like his friend Dennet, who wrote a book brazenly titled Consciousness Explained, Hofstadter believes that he too has explained it. Alas, like Dennet, he has merely described it".

Martin Gardner unlike Tegmark, was a mathematical Platonist rather than a physical one, i.e. he believed that mathematical objects objectively exist out there, but not that the universe is made of them.

There is a fatal flaw with applying Gödel's, or any other mathematical theorem, to philosophy. One has to assume that conditions of the theorem are met in reality, so any consequences a theorem can provide are already baked into the original assumption, and can be painlessly rejected along with it by those who dislike them. I think Wittgenstein had something like that in mind when he said that Gödel's theorem has no philosophical consequences, see Matthíasson's Interpretations of Wittgensteins Remarks on Gödel.